The following is an analysis of confidence interval sizes for Bernoulli Trials. In particular:
How does the width of a 95% confidence interval narrow as the number of trials increases?
The goal is to inform decisions choosing a number of trials \(N\), such that \(N\) is not unnecessarily large, while large enough to give appropriate confidence in estimates resulting from the trials.
The width of the confidence interval is computed as follows:
\(w = f(p, N) = 2z \: \frac{\sqrt{p(1-p)}}{\sqrt{N}} \quad , \quad z = 1.96\)
The following shows the expected width of a 95% Confidence Interval, given the probability of success \(p\), and the number of trials performed \(N\).
The following shows the number of trials (\(N\)) required to narrow the Confidence Interval width to below \(0.05\), for a given probability \(p\). In other words, it shows how the lines in the plot above intersect with \(y = 0.05\).